naginterfaces.library.tsa.uni_​garch_​gjr_​forecast

naginterfaces.library.tsa.uni_garch_gjr_forecast(nt, ip, iq, theta, gamma, ht, et)

uni_garch_gjr_forecast forecasts the conditional variances, $h_t$, for $\text{t}=T+1,\dots ,T+\xi$ from a GJR $\text{GARCH}(p,q)$ sequence, where $\xi$ is the forecast horizon and $T$ is the current time (see Glosten et al. (1993)).

For full information please refer to the NAG Library document for g13ff

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g13/g13fff.html

Parameters

ntint

$\xi$, the forecast horizon.

ipint

The number of coefficients, ${\beta }_{\text{i}}$, for $\text{i}=1,2,\dots ,p$.

iqint

The number of coefficients, ${\alpha }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$.

thetafloat, array-like, shape $(iq+ip+1)$

The first element must contain the coefficient ${\alpha }_o$ and the next $iq$ elements must contain the coefficients ${\alpha }_{\text{i}}$, for $\text{i}=1,2,\dots ,q$. The remaining $ip$ elements must contain the coefficients ${\beta }_{\text{j}}$, for $\text{j}=1,2,\dots ,p$.

gammafloat

The asymmetry parameter $\gamma$ for the $\text{GARCH}(p,q)$ sequence.

htfloat, array-like, shape $\left(\text{num}\right)$

The sequence of past conditional variances for the $\text{GARCH}(p,q)$ process, $h_{\text{t}}$, for $\text{t}=1,2,\dots ,T$.

etfloat, array-like, shape $\left(\text{num}\right)$

The sequence of past residuals for the $\text{GARCH}(p,q)$ process, ${ϵ}_{\text{t}}$, for $\text{t}=1,2,\dots ,T$.

Returns

fhtfloat, ndarray, shape $\left(nt\right)$

The forecast values of the conditional variance, $h_t$, for $\text{t}=T+1,\dots ,T+\xi$.

Raises

NagValueError

(errno $1$)

On entry, $max(ip-1,iq-1)=⟨value⟩$.

Constraint: $max(ip-1,iq-1)\le 20$.

(errno $1$)

On entry, $\text{num}=⟨value⟩$.

Constraint: $max(ip,iq)\le \text{num}$.

(errno $1$)

On entry, $nt=⟨value⟩$.

Constraint: $nt>0$.

(errno $1$)

On entry, $iq=⟨value⟩$.

Constraint: $iq\ge 1$.

(errno $1$)

On entry, $ip=⟨value⟩$.

Constraint: $ip\ge 0$.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

Assume the $\text{GARCH}(p,q)$ process can be represented by:

$$h_t={\alpha }_0+\sum _{i=1}^q({\alpha }_i+\gamma I_{t-i}){ϵ}_{t-i}^2+\sum _{i=1}^p{\beta }_i{h}_{t-i}\text{,}t=1,2,\dots ,T\text{.}$$

where ${ϵ}_t|{\psi }_{t-1}=N(0,h_t)$ or ${ϵ}_t|{\psi }_{t-1}=S_t(\text{df},h_t)$, and $I_t=1$, if ${ϵ}_t<0$, or $I_t=0$, if ${ϵ}_t\ge 0$, has been modelled by uni_garch_gjr_estim(), and the estimated conditional variances and residuals are contained in the arrays $ht$ and $et$ respectively.

uni_garch_gjr_forecast will then use the last $max(p,q)$ elements of the arrays $ht$ and $et$ to estimate the conditional variance forecasts, $h_t|{\psi }_T$, where $t=T+1,\dots ,T+\xi$ and $\xi$ is the forecast horizon.

References

Bollerslev, T, 1986, Generalised autoregressive conditional heteroskedasticity, Journal of Econometrics (31), 307–327

Engle, R, 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica (50), 987–1008

Engle, R and Ng, V, 1993, Measuring and testing the impact of news on volatility, Journal of Finance (48), 1749–1777

Glosten, L, Jagannathan, R and Runkle, D, 1993, Relationship between the expected value and the volatility of nominal excess return on stocks, Journal of Finance (48), 1779–1801

Hamilton, J, 1994, Time Series Analysis, Princeton University Press